>> A utility company burns coal to generate electricity. The cost of removing a certain percent of the pollutants from the smoke stack emission is typically not a linear function. That is, if it cost, let's say C dollars to remove 25 percent of the pollutants, it would cost more than double that, or 2C dollars to remove 50 percent of the pollutants. As the percent of removed pollutants approaches 100 percent, the cost tends to become prohibitive. Suppose that the cost C of removing P percent of the smoke stack pollutants is given by this equation. C equals 80,000 P over 100 minus P, where P is a number between 0 and 100, and it can't be 100. You see, P cannot take on the value of 100 because the denominator would become 0, and therefore, the equation undefined. Alright. Let us sketch the graph of this function. Now, I've already entered the, the equation onto the calculator. It's important to understand that X is playing the part that P plays in the equation. So X corresponds with P, and Y corresponds with C. So X is some percent of pollutants removed, and Y is the cost of removing those pollutants. OK. The window settings. I have looked at this before, and, and so I have made my, my settings accordingly. I set the X max at the point where the, the function is undefined at 100. I backtracked from there. Ninety-four, you know, is the magic number for horizontal distances to have a friendly value when we use trace, and we're going to use trace in this problem. So I backtracked 94 from 100 to get 6 as the X minimum. The, the Y minimum, now, that's a cost figure. Y is the cost figure. I set up the minimum at 0, and the max at one million. And here is the graph. Now, I'll press trace, and let's understand what, what this sort of means, what the X and Y components mean. I want to make sure we understand that before we get into our problem. An X of 60 means that, that we have, we're removing 60 percent of the pollutants, and a Y of 120,000 means that it costs $120,000 to remove 60 percent of the pollutants. Alright. Let's propose a problem. Suppose you're a member of a state legislature, and you're considering a law that would require utility companies to remove 90 percent of the pollutants from their smoke stack emissions. If the current law requires 85 percent removal, how much additional expense would the new law ask the utility company to spend? Well, this is just a matter of tracing to the proper points on our graph, you see. The current law says that we are requiring 85 percent removal of pollutants. So let's see how much that's costing the utility company right now. Oh, it's costing the utility company this much money. Well, that's $453,333 to remove 85 percent of the pollutants. Now, if we want to pass legislation that is asking for 90 percent removal, it's going to cost the utility company $720,000 for that, and the difference between $720,000 and $453,333, you see, would be the additional cost that we are bringing to bear on the utility company, and that difference turns out to be $266,667. So it's, it's a kind of important if we are state legislators to understand what we are going to, to cost that utility company or what this legislation is going to mean in terms of costs to the u, utility company to implement our proposed legislation.